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Problem 03 The picture below shows a fluid of density p₁ and viscosity μ₁ injected at a velocity U through a circular orifice of diameter D into a tank of fluid with density po and viscosity μo = μ₁. The injected fluid forms a cone of angle 0 called the spreading angle. Gravity g acts downward on the two fluids. U 1, με = μο D 0 cm 1 2 3 4 5 6 7 8 9 ρο, μο Ꮎ R(z) u(z) א (a) Provide the dimensions and the SI units of all of these quantities: po, P1, μo, U, D, g and 0. (b) Using the Buckingham-Pi theorem, determine the dimensionless groups that control 0. 1.0 × 103, p1 po = 1000 and g = 9.8 are given in the SI units. Compute the values of the dimensionless groups for this experiment. The spreading angle can be measured from the figure. (c) In the experiment pictured above, D = 0.002, U = 7.5, μo = (d) Under certain hypotheses, mass and momentum conservation lead to the following equation: du 5 R²u +tan(0) Ru² dz 6 μα Po dz2 (R²u) P1-PO R²g, ρο where u represents the axial velocity averaged on a cross-section of the cone, R represents the cone radius at an axial distance z and g is the gravitational acceleration. Rewrite this equation in a non-dimensional form, where you will show the Pi groups found in the previous question.

Problem 03 The picture below shows a fluid of density p₁ and viscosity μ₁ injected at a velocity U through a circular orifice of diameter D into a tank of fluid with density po and viscosity μo = μ₁. The injected fluid forms a cone of angle 0 called the spreading angle. Gravity g acts downward on the two fluids. U 1, με = μο D 0 cm 1 2 3 4 5 6 7 8 9 ρο, μο Ꮎ R(z) u(z) א (a) Provide the dimensions and the SI units of all of these quantities: po, P1, μo, U, D, g and 0. (b) Using the Buckingham-Pi theorem, determine the dimensionless groups that control 0. 1.0 × 103, p1 po = 1000 and g = 9.8 are given in the SI units. Compute the values of the dimensionless groups for this experiment. The spreading angle can be measured from the figure. (c) In the experiment pictured above, D = 0.002, U = 7.5, μo = (d) Under certain hypotheses, mass and momentum conservation lead to the following equation: du 5 R²u +tan(0) Ru² dz 6 μα Po dz2 (R²u) P1-PO R²g, ρο where u represents the axial velocity averaged on a cross-section of the cone, R represents the cone radius at an axial distance z and g is the gravitational acceleration. Rewrite this equation in a non-dimensional form, where you will show the Pi groups found in the previous question.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Problem 03 The picture below shows a fluid of density p₁ and viscosity μ₁ injected at a velocity U through a circular orifice of diameter D into a tank of fluid with density po and viscosity μo = μ₁. The injected fluid forms a cone of angle 0 called the spreading angle. Gravity g acts downward on the two fluids. U 1, με = μο D 0 cm 1 2 3 4 5 6 7 8 9 ρο, μο Ꮎ R(z) u(z) א (a) Provide the dimensions and the SI units of all of these quantities: po, P1, μo, U, D, g and 0. (b) Using the Buckingham-Pi theorem, determine the dimensionless groups that control 0. 1.0 × 103, p1 po = 1000 and g = 9.8 are given in the SI units. Compute the values of the dimensionless groups for this experiment. The spreading angle can be measured from the figure. (c) In the experiment pictured above, D = 0.002, U = 7.5, μo = (d) Under certain hypotheses, mass and momentum conservation lead to the following equation: du 5 R²u +tan(0) Ru² dz 6 μα Po dz2 (R²u) P1-PO R²g, ρο where u represents the axial velocity averaged on a cross-section of the cone, R represents the cone radius at an axial distance z and g is the gravitational acceleration. Rewrite this equation in a non-dimensional form, where you will show the Pi groups found in the previous question.
Transcribed Image Text:Problem 03 The picture below shows a fluid of density p₁ and viscosity μ₁ injected at a velocity U through a circular orifice of diameter D into a tank of fluid with density po and viscosity μo = μ₁. The injected fluid forms a cone of angle 0 called the spreading angle. Gravity g acts downward on the two fluids. U 1, με = μο D 0 cm 1 2 3 4 5 6 7 8 9 ρο, μο Ꮎ R(z) u(z) א (a) Provide the dimensions and the SI units of all of these quantities: po, P1, μo, U, D, g and 0. (b) Using the Buckingham-Pi theorem, determine the dimensionless groups that control 0. 1.0 × 103, p1 po = 1000 and g = 9.8 are given in the SI units. Compute the values of the dimensionless groups for this experiment. The spreading angle can be measured from the figure. (c) In the experiment pictured above, D = 0.002, U = 7.5, μo = (d) Under certain hypotheses, mass and momentum conservation lead to the following equation: du 5 R²u +tan(0) Ru² dz 6 μα Po dz2 (R²u) P1-PO R²g, ρο where u represents the axial velocity averaged on a cross-section of the cone, R represents the cone radius at an axial distance z and g is the gravitational acceleration. Rewrite this equation in a non-dimensional form, where you will show the Pi groups found in the previous question.
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